Laplacian with singular potential

Question

Let $S$ be a $2$-dimensional sphere. Let $p$ be a point in $S$. Let $L$ be a second order elliptic partial differential operator with smooth coefficients defined over the complement of $p$. Near $p$, $S$ is parametrised conformally by the Poincaré Disk, $\Bbb{D}^*$. We suppose that, with respect to this parametrisation, $L$ has the form

$$ L = \Delta + \alpha r^{-2} + \beta \log(r)^2 r^{-2}, $$

where $\Delta$ is the standard Laplacian and $r=\|x\|$.

Can someone please tell me which function spaces I should use over $S$ to make $L$ into a Fredholm operator, and what the Fredholm Index of the resulting operator would be?

Thanks.

Answer 1

When $\alpha=0$ and $\beta<0$, which is the case which really interests me, it turns out to actually be a fairly straightforward application of the theory of unbounded operators. First, to simplify things, we suppose that $\beta=-1$, and we compose with the conformal mapping $\Phi:S^1\times[0,\infty[\rightarrow\Bbb{D}^*$ given by $\Phi(\theta,t)=e^{i\theta-t}$. This leads to the operator

$$ L:= \Delta - t^2 = \partial_\theta^2 + \partial_t^2 - t^2. $$

We define the graph of $L$, $G(L)$ to be the set of all pairs $(f,g)\in L^2\times L^2$ such that $Lf=g$ in the weak sense. Since this is a closed condition, the graph of $L$ is a closed subspace of $L^2\times L^2$ and is therefore a Hilbert space in its own right. Furthermore, the projection onto the first component is injective, and we define the domain of $L$, $D(L)$ to be the image of this projection. We furnish the domain of $L$ with the unique norm that makes this projection into an isometry. That is, for all $f\in D(L)$

$$ \|f\|_{D(L)}^2 = \|f\|_{L^2}^2 + \|Lf\|_{L^2}^2. $$

Using the negativity of the potiential, we show that the canonical embeding of $D(L)$ into $L^2$ is a compact mapping. From this, we deduce that $L$ has finite dimensional kernel (actually, trivial kernel), and closed image. We then show that the image of $L$ contains every smooth function of compact support. It follows from closure of the image that $L$ is surjective. In summary, $L$ defines a linear isomorphism from $D(L)$ into $L^2$. [1] and [2] are good references for these standard techniques.

Finally, with a little bit more work, we show that the norm $\|\cdot\|_{D(L)}$ is equivalent to the following weighted Sobolev Norm:

$$ \|f\|_{H^2_w}^2 = \int t^2 f^2 + \int t \|Df\|^2 + \int \left|\Delta f\right|^2. $$

$\|\cdot\|_{H^2_w}$ is therefore the desired norm.

[1] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order.

[2] Lax P.D., Functional Analysis.

Bernard Helffer's courses on Spectral Theory are probably also helpful.

http://www.math.u-psud.fr/~helffer/